Nearest Neighbor Methods for Non-Euclidean Manifolds

ABSTRACT

Embodiments of the invention disclose a system and a method for determining a nearest neighbor to an input data point on a non-Euclidean manifold. The data points on the non-Euclidean manifold are clustered, projected into Euclidean sub-space nearest to the cluster and mapped from the Euclidean sub-space into a Hamming space such that neighboring data points of the Hamming space corresponds to neighboring data points on the non-Euclidean manifold. The method maps the input data point to the Hamming space corresponding to a particular Euclidean sub-space, wherein the particular Euclidean sub-space is the nearest to the input data point, and selects a data point corresponding to a nearest data point to the input data point in the Hamming space as the nearest neighbor for the input data point on the non-Euclidean manifold.

FIELD OF THE INVENTION

The present invention relates generally to determining a nearestneighbor of a data point on a manifold, and more particularly todetermining a nearest neighbor for a data point on a non-Euclideanmanifold.

BACKGROUND OF THE INVENTION

Human action analysis using computer vision techniques enablesapplications such as automatic surveillance, behavior analysis, andelderly care. However, the automatic analysis of human motion in videosis currently limited to relatively simple classes of predefined motions,small data sets and simple human actions, such as a single personperforming a single primitive action, in a video that is relativelyshort in length.

In contrast, in a real-life surveillance scenario, video data are oftencontinuously recorded and saved for later analysis. In a typical case, asearch for a specific instance of an activity in the video data canresult in days of watching video to find images of interest. Performingsemantic queries such as “find all instances where a person is walkingfrom left to right”, or “find instances where a person starts walkingand then starts running” remains very difficult.

Approximate Nearest Neighbor

Approximate nearest neighbor (ANN) methods, such as variants of localitysensitive hashing (LSH), semantic hashing, and spectral hashing, arecomputationally efficient for finding objects similar to a query objectin large datasets. Those methods have been used to quickly search imagesin web-scale datasets that can contain millions of images.Unfortunately, the key assumption in those methods is that data pointsin the dataset are in a Euclidean space and can only be compared usingEuclidean distances.

This assumption is not always valid and poses a challenge to severalcomputer vision applications where data commonly are defined on complexnon-Euclidean manifolds. In particular, dynamic data, such as humanactivities, are usually represented as dynamical systems, which lie onnon-Euclidean manifolds. Accordingly, the search for the nearestneighbor of the data point has to consider the geometry of the manifold.

Spectral Hashing

As shown in FIG. 2, a spectral hashing (SH) method is an example ofhashing methods that map data points lying on Euclidean manifold 210onto Hamming space 220 such that neighboring data points 225 in Hammingspace correspond to neighboring data points 215 on the Euclideanmanifold.

Accordingly, for the data points,

{x_(i)}_(i=1) ^(N)εR^(d),

the goal of the spectral hashing is to find k-bit binary vectors,

{y _(i)}_(i=1) ^(N)ε{−1,1}^(k)

such that similar points in, R^(d) under the similarity measure,

$W_{ij} = {\exp( {- \frac{{Px}_{i} - {x_{j}P^{2}}}{ɛ^{2}}} )}$

and map to binary vectors that are close to each other under the Hammingdistance weighted by a weighting function W. If the data points X_(i)are sampled from a probability distribution p(x), then the SH solves thefollowing optimization problem:

minimize ∫∥y(x₁)−y(x₂)∥²W(x₁,x₂)

p(x₁)p(x₂)dx₁dx₂

s.t. y(x)ε{−1,1}^(k)

∫y(x)p(x)dx=0

∫y(x)y(x)^(T) p(x)dx=I  (1)

Relaxing the first constraint gives a solution y for the Equation (1) asthe first k eigenfunctions of the weighted Laplace-Beltrami operator onthe manifold. If the distribution p is multi-dimensional uniformdistribution on the Euclidean space R^(d) and the weighting function Wis defined as above, then there is one closed form solution for theseeigenfunctions.

If the distribution p is a Gaussian distribution on the Euclidean spaceR^(d), there exists an iterative solution.

The spectral hashing method is summarized into the following steps:

Determining principal components of data using principal componentanalysis (PCA);Compute the k smallest single-dimension analytical eigenfunctions of theLaplace-Beltrami operator under the specified weighting function andprobability distribution by using a rectangular approximation alongevery PCA direction; andThreshold the analytical eigenfunctions computed for each data point atzero, to obtain binary codes.

In theory, any probability distribution on a general manifold and aweighting function can be used to analytically compute theeigenfunctions of the corresponding Laplace-Beltrami operator. However,even for scalar Euclidean data, such computation remains an open andunsolved problem.

In the case of non-Euclidean data that for example represent humanactivities, such an analysis becomes extremely difficult. Thedistribution of the data points is usually unknown, and even if a formof the distribution is assumed, a closed-form representation for thedistribution on a particular manifold might not exist. Moreover, theweighting function is no longer a simple exponential similarity functionas the function is based on geodesic or chord distances on the manifold.Finally, the exact computation of the solution of the minimizationproblem in Equation (1) for any general weighting function, probabilitydistribution on any arbitrary manifold is extremely difficult.

Kernel Spectral Hashing (KSH) method uses kernel PCA instead of PCA tofind the eigenfunctions. The method embeds the data points in ahigh-dimensional Euclidean space, and finds the value of theeigenfunction at each data point. However, the KSH method computes thekernel of an input data point with all the data points in a training setthat used to compute the kernel PCA components. This is ascomputationally complex as performing exact nearest neighbors by usingthe kernel as an affinity measure. Even though a well-chosen kernelmight give very good results in terms of retrieval accuracy, the KSHmethod has a computational complexity of O(N), where N is the number ofthe data points in the training set, which could be in the millions.

Accordingly, it is desired to provide an efficient method fordetermining the nearest neighbor for the data points lying on anon-Euclidean manifold.

SUMMARY OF THE INVENTION

Embodiments of the invention are based on a realization that approximatenearest-neighbor methods, e.g., a spectral hashing, can not be useddirectly for data points on a non-Euclidean manifold, and projecting theentire data set into Euclidean space results in large distortions ofintrinsic distances.

Therefore, the embodiments of the invention cluster the data points onthe non-Euclidean manifold into a set of clusters. For each cluster, anearest Euclidean sub-space forming a set of Euclidean sub-spaces isdetermined.

The data points of each cluster are projected into correspondingEuclidean sub-space, such that each cluster is approximated by datapoints lying on the Euclidean sub-space to produce a set of approximatedclusters.

Each of the approximated clusters is mapped into a corresponding Hammingspace to produce a set of Hamming clusters, such that neighboring datapoints on the Hamming cluster corresponds to neighboring data points onthe non-Euclidean manifold.

One embodiment of the invention discloses a method for determining anearest neighbor to an input data point lying on a non-Euclideanmanifold from data points lying on the non-Euclidean manifold. Themethod clusters the data points into a set of clusters; determines, foreach cluster, a Euclidean sub-space nearest to the cluster to form a setof Euclidean sub-spaces; projects the data points of each cluster intothe Euclidean sub-space nearest to the cluster, such that each clusteris approximated by data points lying on the Euclidean sub-space toproduce a set of approximated clusters; maps each of the approximatedclusters into a corresponding Hamming space to produce a set of Hammingclusters, such that neighboring data points of the Hamming clustercorresponds to neighboring data points on the non-Euclidean manifold;maps the input data point to the Hamming cluster corresponding to aparticular Euclidean sub-space, wherein the particular Euclideansub-space is nearest to the input data point; and selects the data pointcorresponding to the nearest data point to the input data point in theHamming space as the nearest neighbor for the input data point on thenon-Euclidean manifold, wherein the steps are performed in a processor.

Another embodiment discloses a method for determining a nearest neighborto an input data point on a non-Euclidean manifold. The data points onthe non-Euclidean manifold are clustered, projected into Euclideansub-space nearest to the cluster and mapped from the Euclidean sub-spaceinto a Hamming space such that neighboring data points of the Hammingspace corresponds to neighboring data points on the non-Euclideanmanifold. The method maps the input data point to the Hamming spacecorresponding to a particular Euclidean sub-space, wherein theparticular Euclidean sub-space is the nearest to the input data point,and selects a data point corresponding to a nearest data point to theinput data point in the Hamming space as the nearest neighbor for theinput data point on the non-Euclidean manifold.

Yet another embodiment discloses a system for determining a nearestneighbor to an input data point lying on a non-Euclidean manifold fromdata points lying on the non-Euclidean manifold, wherein the data pointsare clustered and the data points of each cluster are projected intoEuclidean sub-space nearest to the cluster and mapped from the Euclideansub-space into a Hamming space producing a set of Hamming spaces suchthat neighboring data points of the Hamming space corresponds toneighboring data points on the non-Euclidean manifold. The systemcomprises a processor configured to map the input data point to theHamming space corresponding to a particular Euclidean sub-space, whereinthe particular Euclidean sub-space is the nearest to the input datapoint, wherein the mapping is performed by a processor; and means forselecting a data point corresponding to a nearest data point to theinput data point in the Hamming space as the nearest neighbor for theinput data point on the non-Euclidean manifold.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure is a block diagram of a method for determining the nearestneighbor for an input data point lying on non-Euclidean manifoldaccording to embodiments of invention;

FIG. 2 is a schematic of a prior art spectral hashing method;

FIG. 3 is schematic of embodiment, wherein the non-Euclidean manifold isRiemannian manifold;

FIG. 4 is a block diagram of Riemannian k-means method for clusteringdata points according to one embodiment of the invention; and

FIG. 5 is a block diagram of a method for determining representativedata points of clusters according to another embodiment of theinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Embodiments of the invention reduce a task of determining a nearestneighbor for an input data point on non-Euclidean manifold todetermining which particular Euclidean sub-space from the set ofEuclidean sub-spaces is the nearest to the input data point, projectingthe input data point into the particular Euclidean sub-space, andmapping the projection onto the corresponding Hamming space. A datapoint corresponding to the nearest data point to the input data point inthe Hamming space is selected as the nearest neighbor for the input datapoint on the non-Euclidean manifold.

FIG. 1 shows a block diagram for a method 100 for determining a nearestneighbor 190 for an input data point 105 lying on the non-Euclideanmanifold. Data points 110 lying on non-Euclidean manifold are clustered120 on the manifold into a set of K clusters 125. The steps of themethod can be performed in a processor 101 including memory andinput/output interfaces as known in the art.

For each cluster in a set 125, the nearest Euclidean sub-space, e.g.,tangent space, is determined 130 to form a set of Euclidean sub-spaces135, and the data points of each cluster are projected 140 intocorresponding nearest Euclidean sub-space, e.g., a nearest Euclideansub-space 136, producing a set of approximated clusters 145.

Each approximated cluster 145 approximates the data points of eachcluster 125 on the corresponding nearest Euclidean sub-space.

Then, each approximated cluster is mapped 150 separately onto acorresponding Hamming space using, e.g., the spectral hashing (SH)method described above, producing a set of Hamming clusters 155, suchthat neighboring data points on a Hamming cluster from the set 155correspond to neighboring data points on the non-Euclidean manifold 110.

The method 100 determines 160, which Euclidean sub-space from the set ofEuclidean sub-spaces 136 is the nearest to the input data point 105. Forexample, in one embodiment, the nearest sub-space 136 is selected as theEuclidean sub-space corresponding to a cluster having a center at aminimum geodesic distance from the input data point.

In another embodiment, the closest sub-space is selected as theEuclidean sub-space having a minimal reconstruction error for the inputdata point.

The input data point is projected on the nearest sub-space and mapped tothe Hamming space as described above. A data point 170 corresponding toa data point 165 having a minimal Hamming distance to the input datapoint in the Hamming space is selected as the nearest neighbor 190 forthe input data point on the non-Euclidean manifold.

Since the clustering requires only K distance evaluations, e.g.,geodesic distances, or a reconstruction error, a computational cost ofthe method 100 is O(K) rather than O(N) as in the prior art KSH methoddescribed above, where K<<N. Moreover, the clustering of the data pointsbetter approximates the uniform distribution assumption in each cluster.

Riemannian Spectral Hashing (RSH)

FIG. 3 shows an embodiment, wherein the data points 311 lie onRiemannian manifold 310, and the Euclidean sub-spaces 321-322 aretangent spaces. The tangent space T_(y) M to a manifold, at a point y isa Euclidean space. Therefore, the data points {x_(i)}¹⁻¹ ^(N) on themanifold to the tangent space at the data point y, i.e., the point 330,can be projected by using, e.g., a logarithm mapping, Δ_(i)={right arrowover (yx)}_(i)=log_(y)(x_(i)), and the spectral hashing can be performedon the tangent space projections, {Δ_(i)}_(i=1) ^(N).

Accordingly, one embodiment of the invention clusters the data pointsinto K clusters and selects a center of each cluster as the pole of thecorresponding tangent space.

FIG. 4 shows one variation of this embodiment, which uses Riemanniank-means method for clustering the data points according to the followingsteps:

Initialize 410 cluster centers {c_(j)}_(j=1) ^(K) by randomly selectingK points from the data points 460.For each data point x_(i), determine 420 geodesic distance to eachcluster center according to

d(c _(j) ,x _(i))=∥log_(c) _(j) (x _(i))∥.

Assign 430 the cluster center nearest to the data point as a clustermembership w of the data point according to

w _(i)=argmin_(j)∥log_(c) _(j) (x _(i))∥

Recompute 440 each cluster center as an extrinsic mean of the datapoints in each cluster based on the cluster membership according to

c _(j)=mean{x _(l) |w _(l) =j}

Repeat 470 steps 3 and 4 until convergence, which can require repeateduses of the exponential map and the logarithm map on the manifold untilconvergence to the extrinsic mean.

After the cluster centers and the cluster memberships 450 aredetermined, all data points in the same cluster are projected to thetangent space around the cluster center using the correspondinglogarithm maps. A separate spectral hashing method is then applied 380on each tangent space to map the approximated clusters to the Hammingspace 370.

The input data point z 340 is mapped to the Hamming space by firstdetermining the geodesic distances 350 of the input data point with allthe cluster centers and project 360 the input data point to the tangentspace of the nearest cluster center C_(k), wherein k is determinedaccording to

k=arg_(min) _(j) ∥{right arrow over (c_(j)z)}∥.

Then, the hashing method, e.g., the spectral hashing described above, isused to map the input data point projected on the tangent space to theHamming space according to Δ_(z)=log_(c) _(k) (z).

Distributed Kernel Spectral Hashing (DKSH)

It is not always possible to perform Riemannian k-means method on amanifold. For example, when the logarithm and the exponential maps arenot closed-form or are not defined because of the complexity of themanifold, the cluster centers cannot be determined.

Accordingly, one embodiment of the invention uses a non-lineardimensionality reduction method, such as multidimensional scaling (MDS),to project the data points into a low-dimensional Euclidean space andperforms k-means method on this low-dimensional space.

In different variation of this embodiment, non-linear clusteringmethods, such as kernel k-means or spectral clustering, are used todetermine cluster associations of the data points. Accordingly, insteadof cluster centers, only cluster associations for the data points aredetermined. After the clustering, one representative data point isselected in each cluster to represent the cluster.

FIG. 5 shows a method for determining the representative data points ofthe clusters. The method determines 510 an N×N affinity matrix W 516 ofthe data points 560 based on a kernel 515, or affinity defined on themanifold. MDS 520 is executed using the affinity matrix to produce alow-dimensional Euclidean representation {u_(i)}_(i=1) ^(N) 525 andk-means method 530 is executed on the data points in the low-dimensionalspace determining K cluster centers {v_(j)}_(j=1) ^(K) 535 in thelow-dimensional space.

For each cluster center v_(j), select 540 data point u_(j) 545 nearestto each cluster center in the low-dimensional space, and determine 550the representative data points {x_(p;j)}_(j=1) ^(K) 555 on the manifoldcorresponding to the mapped data points {v_(j)}_(j=1) ^(K) after MDS.

After the representative data point for each cluster has beendetermined, the kernel spectral hashing (KSH) 570 is separately trainedfor each cluster to map 575 the data points to the Hamming space.

Similarly to the RSH method, the input data point z is mapped to theHamming space by determining an affinity W(x_(p;j),z) of the input datapoint with each representative data point, selecting a cluster j havinghighest affinity with the input data point, and using the KSH of thatcluster to map the input data point to the Hamming space to retrieve thenearest neighbors.

The overall complexity of the method according this embodiment isapproximately O(K+N/K), which is more computationally expensive then theRSH method, but significantly better than the complexity of the KSHmethod.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications may be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

1. A method for determining a nearest neighbor to an input data pointlying on a non-Euclidean manifold from data points lying on thenon-Euclidean manifold, comprising the steps of: clustering the datapoints into a set of clusters; determining, for each cluster, aEuclidean sub-space nearest to the cluster to form a set of Euclideansub-spaces; projecting the data points of each cluster into theEuclidean sub-space nearest to the cluster, such that each cluster isapproximated by data points lying on the Euclidean sub-space to producea set of approximated clusters; mapping each of the approximatedclusters into a corresponding Hamming space to produce a set of Hammingclusters, such that neighboring data points of the Hamming clustercorresponds to neighboring data points on the non-Euclidean manifold;mapping the input data point to the Hamming cluster corresponding to aparticular Euclidean sub-space, wherein the particular Euclideansub-space is nearest to the input data point; and selecting the datapoint corresponding to the nearest data point to the input data point inthe Hamming space as the nearest neighbor for the input data point onthe non-Euclidean manifold, wherein the steps are performed in aprocessor.
 2. The method of claim 1, wherein the Euclidean sub-spacenearest to the cluster is a tangent space.
 3. The method of claim 1,further comprising: mapping the approximated cluster to thecorresponding Hamming space using a spectral hashing method.
 4. Themethod of claim 1, further comprising: determining geodesic distancesbetween the input data point and centers of each cluster; and selectingthe particular Euclidean sub-space as the Euclidean sub-spacecorresponding to the cluster having a center at a minimum geodesicdistance from the input data point.
 5. The method of claim 1, furthercomprising: selecting the particular Euclidean sub-space as theEuclidean sub-space having a minimal reconstruction error for the inputdata point.
 6. The method of claim 1, wherein the data points lie on aRiemannian manifold, and the set of Euclidean sub-spaces is a set oftangent spaces.
 7. The method of claim 1, wherein the clustering furthercomprises: clustering the data points based on a Riemannian k-meansmethod.
 8. The method of claim 1, wherein the clustering furthercomprises: initializing cluster centers by randomly selecting K pointsfrom the data points; determining, for each data point, a geodesicdistance to each cluster center; assigning the cluster center nearest tothe data point as a cluster membership of the data point; recomputingeach cluster center as an extrinsic mean of the data points in eachcluster based on the cluster membership; and repeating the assigning andrecomputing until convergence to the extrinsic mean.
 9. The method ofclaim 1, further comprising: projecting the data points into alow-dimensional Euclidean space using a non-linear dimensionalityreduction; and determining a cluster association of the data points onthe low-dimensional Euclidean space.
 10. The method of claim 9, furthercomprising: performing the non-linear dimensionality reduction based onmultidimensional scaling.
 11. The method of claim 9, further comprising:performing a non-linear clustering for the determining the clusterassociation based on kernel k-means and/or spectral clustering.
 12. Themethod of claim 9, further comprising: selecting a representative datapoint in each cluster to represent the cluster.
 13. The method of claim9, wherein the selecting further comprises: determining a mapped datapoint nearest to a center of the cluster in the low-dimensionalEuclidean space; and selecting the representative data point on themanifold corresponding to the mapped data point.
 14. The method of claim12, further comprising: mapping the data points of each cluster to theHamming space based on kernel spectral hashing (KSH).
 15. A method fordetermining a nearest neighbor to an input data point lying on anon-Euclidean manifold from data points lying on the non-Euclideanmanifold, wherein the data points are clustered and the data points ofeach cluster are projected into Euclidean sub-space nearest to thecluster and mapped from the Euclidean sub-space into a Hamming spaceproducing a set of Hamming spaces such that neighboring data points ofthe Hamming space corresponds to neighboring data points on thenon-Euclidean manifold, comprising steps of: mapping the input datapoint to the Hamming space corresponding to a particular Euclideansub-space, wherein the particular Euclidean sub-space is the nearest tothe input data point, wherein the mapping is performed by a processor;and selecting a data point corresponding to a nearest data point to theinput data point in the Hamming space as the nearest neighbor for theinput data point on the non-Euclidean manifold.
 16. The method of claim15, further comprising: determining geodesic distances between the inputdata point and centers of each cluster; and selecting the particularEuclidean sub-space as the Euclidean sub-space corresponding to acluster having a center at a minimum geodesic distance from the inputdata point.
 17. The method of claim 15, further comprising: selectingthe particular Euclidean sub-space as the Euclidean sub-space having aminimal reconstruction error for the input data point.
 18. A system fordetermining a nearest neighbor to an input data point lying on anon-Euclidean manifold from data points lying on the non-Euclideanmanifold, wherein the data points are clustered and the data points ofeach cluster are projected into Euclidean sub-space nearest to thecluster and mapped from the Euclidean sub-space into a Hamming spaceproducing a set of Hamming spaces such that neighboring data points ofthe Hamming space corresponds to neighboring data points on thenon-Euclidean manifold, comprising: a processor configured to map theinput data point to the Hamming space corresponding to a particularEuclidean sub-space, wherein the particular Euclidean sub-space is thenearest to the input data point, wherein the mapping is performed by aprocessor; and means for selecting a data point corresponding to anearest data point to the input data point in the Hamming space as thenearest neighbor for the input data point on the non-Euclidean manifold.19. The system of claim 18, further comprising: means for determininggeodesic distances between the input data point and centers of eachcluster; and means for selecting the particular Euclidean sub-space asthe Euclidean sub-space corresponding to a cluster having a center at aminimum geodesic distance from the input data point.
 20. The system ofclaim 18, further comprising: means for selecting the particularEuclidean sub-space as the Euclidean sub-space having a minimalreconstruction error for the input data point.